I. TWO WAYS THAT A C-INFINITY FUNCTION CAN FAIL TO BE ANALYTICII. EARLY HISTORY OF NOWHERE ANALYTIC C-INFINITY FUNCTIONSIII. ZAHORSKI'S 1947 CHARACTERIZATIONIV. MOST C-INFINITY FUNCTIONS ARE NOWHERE ANALYTICV. UNEXPLORED AREAS AND OTHER RESULTSVI. REFERENCES

Every C-infinity function has a formal Taylor series expansionabout each point. If this Taylor expansion converges to theoriginal function in neighborhood of x=b, then the function issaid to be analytic at x=b. There are two ways this can fail:

(C) The Taylor expansion converges in a neighborhood of x=b, but in no neighborhood of x=b does the Taylor expansion converge to the given function.

(P) The Taylor expansion fails to converge in every neighborhood of x=b (i.e. the Taylor series at x=b has a zero radius of convergence).

Following Zahorski [35], we classify the non-analytic pointsof a C-infinity function into the following two categories.A point of non-analyticity is called a (C)-point (for Cauchy)if it belongs to (C) above. A point of non-analyticity is calleda (P)-point (for Pringsheim) if it belongs to (P) above.

An example of a (C)-point is given at x=0 by

exp(-1 / x^2) [with f(0) = 0].

An example of a (P)-point is given at x=0 by

Integral(t=0 to infinity) of [exp(-t) * cos(x*t^2)] dt.

The first example is well known. The second example is discussedon pp. 189-190 of Boas [5]. Cauchy (1823) was the first to givean example of a (C)-point (the same example given above),while Du Bois-Reymond (1876) was the first to give an exampleof a (P)-point. Du Bois-Reymond's example is

SUM(n=1 to infinity) of (-1)^(n+1) * x^(2n) / [(2n)! * (x^2 + b_n)],

where b_n = (a_n)^2 and {a_n} is any sequence of nonzero realnumbers whose limit is zero.

Du Bois-Reymond's 1876 paper appeared in a journal that was notvery well known [Enter "08.0127.01" (without quotes) in the"JFM. no." window at .],and so his example was not generally known at that time.Du Bois-Reymond rewrote and expanded his 1876 paper and publishedthe revision in the much better known journal "MathematischeAnnalen". His revised paper, published in 1883, can be obtainedon the internet at

Du Bois-Reymond's example is given at the bottom of page 111 ofhis 1883 paper. Incidentally, there are some minor gaps in hisproof that the Taylor expansion diverges for nonzero values ofx that arise from overzealous manipulations of divergent series.

Even in the late 1800's it was well known that the set ofnon-analytic points of any function is a closed set. This is simplya restatement of the fact that if a function is analytic at somepoint, then it is analytic in a neighborhood of that point(i.e. the points of analyticity form an open set). Therefore, inorder to obtain a nowhere analytic C-infinity function, it isonly necessary to find a C-infinity function having a dense setof non-analytic points.

In 1895 Borel described a simple way to obtain nowhere analyticC-infinity functions. Let f(z) be a complex-valued function of acomplex variable z that is analytic in an open disk and C-infinityon the closure of this disk, and which cannot be continued analytically across the boundary of this disk. Then f(ix), wherex is a real variable, is a complex-valued function that isC-infinity and nowhere analytic in the interval -Pi < x < Pi.[If you want a real-valued example, take either the real part orthe imaginary part of this function.]

A simple example of such a function f(z) was given in 1891 byFredholm:

For more information on the early history of nowhere analyticC-infinity functions, see Bilodeau [3]. For a general survey ofnowhere analytic C-infinity functions, including issues relatingto Zahorski's theorem discussed below, see Salzmann/Zeller [26].An elementary verification of a nowhere analytic C-infinity functionis given in Merryfield [21], a paper that you can download at

None of these early examples was shown to have a (C)-pointeverywhere or to have a (P)-point everywhere. Boas provedin 1935 (see [4] or p. 192 of [5]) that the (P)-pointsof a nowhere analytic C-infinity function are dense in R.Therefore, it is not possible to have an example in which everypoint is a (C)-point. However, the possibility that every pointcould be a (P)-point remained open. The first example of aC-infinity function such that every point is a (P)-point wasgiven by Cartan in 1940 ([6], pp. 20-22). In 1947 Zahorski [35]gave the following characterization for the non-analytic pointsof a C-infinity function. [An announcement [34] of this result wasmade in 1946.]

THEOREM (Zahorski): A necessary and sufficient condition for two sets of real numbers C and P to be the (C)-points and the (P)-points, respectively, of some C-infinity function is that the following four properties hold:

(a) C is a first category F_sigma set. (b) P is a G_delta set. (c) C is disjoint from P. (d) C union P is closed in R.

[[ Zahorski died on May 8, 1998 at the age of 84. I believe his work on the singularities of C-infinity functions was part of his Ph.D. research. ]]

We mention three corollaries.

** If P is empty in some interval, then the set of (C)-points in that interval is a relatively closed first category subset of that interval. Since every closed first category set is nowhere dense, no such function can be nowhere analytic in that interval. Hence, if a function is nowhere analytic and C-infinity, then every interval must contain some (P)-points, and we get the result that Boas proved in 1935.

** Another corollary is the existence of a C-infinity function having a (P)-point everywhere: Choose C to be empty and P = R.

** Still another corollary is that there exist C-infinity functions belonging to each of the following categories (recall Pringsheim's examples in Section II):

(i) C is c-dense in R and P is empty. (ii) C is empty and P is c-dense in R. (iii) Both C and P are c-dense in R.

The Baire category theorem can be used to prove the existence ofnowhere analytic C-infinity functions, and this has beenre-discovered several times. Using the standard metric onC-infinity (more precisely, any metric that generates the topologyof uniform convergence for all orders of derivatives on compactsets), Morgenstern [22] (1954) gave a concise proof that theBaire-typical C-infinity function is nowhere analytic. An outlineof a proof can be found on pp. 301-302 of Dugundji [12], withMorgenstern's name mentioned. However, Dungundji states thatthe particular proof he presents is due to Salzmann and Zeller,apparently from a personal communication. For an expanded versionin English of Morgenstern's original proof, see pp. 95-97 ofJones [16].

Next up, we have Christensen [9] (1972), who proves the sameresult, unaware of Morgenstern. Then we have Darst [10] (1973),who was apparently unaware of both Morgenstern's and Christensen'spapers. However, Darst followed this up with [11] (1974), where a farstronger Baire-typical result is proved. [See page 26. The resultI am alluding to only shows up in the proof of a certain theorem,not in any of his theorem statements.] Darst shows that certainquasi-analytic classes have a Baire-typical set of functions thatare nowhere quasi-analytic relative to other quasi-analytic notions.Next, we have Cater [7] (1984), who was aware of both Christensen'sand Darst's papers, and probably also of Morgenstern's paper.[Darst writes (p. 618): "Two references in English are ...".]Cater's paper is well written and his proof has all the detailsworked out. Siciak [29] (1986), who was aware of Morgenstern'spaper, proves a Baire-typical multi-dimensional analog ofMorgenstern's result (theorem 10 on p. 144). After this, thereis Bernal [2] (1987). Bernal states that a corollary of the mainresult in his paper is that the Baire-typical C-infinity functionis nowhere analytic. [Bernal cites Cater, Christensen, and Darstin his bibliography.] Finally, Ramsamujh [23] (1991) proves thatthe Baire-typical C-infinity function is nowhere analytic, unawareat that time of any of the preceding papers.

Aside from simply being nowhere analytic, can we say anything aboutthe (C)-points and the (P)-points of the Baire-typical C-infinityfunction? As far as I can tell, it appears that all the sources Imentioned in the previous two paragraphs, except for Bernal [2],Ramsamujh [23], and Siciak [29], prove only that the Baire-typicalC-infinity function has a dense set of (P)-points. From Zahorski'stheorem we know that the set of (P)-points is a G_delta set(actually, this is immediate from the definition), and so we havethe interesting observation that the Baire-typical C-infinityfunction has a Baire-typical set of (P)-points. In fact,Christensen [9] explicitly states his result in this way.

However, Bernal [2], Ramsamujh [23], and Siciak [29] manage toprove more. They actually prove that EVERY point is a (P)-pointfor the Baire-typical C-infinity function. Thus, the Baire-typicalC-infinity function has no (C)-points at all. Although thiscompletely settles the matter, the result is unfortunate becauseit closes the door on the possibility of investigating the Lebesguemeasure (or Hausdorff dimension, if measure zero) of the sets of(C)-points and (P)-points of the Baire-typical C-infinity function.

If we let f(x) = exp(-1 / x^2) for x > 0 and zero elsewhere, then fis a C-infinity function whose Taylor series at x=0 converges to fin a left neighborhood of x=0 but doesn't converge to f in anyright neighborhood of x=0. Let's call such a point for a functiona "right (P)-point" of that function, and define "left (P)-point"of a function in the obvious way. Finally, we say a point is a"bilateral (P)-point" of a function if the Taylor series for thefunction about that point has a positive radius of convergence,but it fails to converge to the function in every left neighborhoodand every right neighborhood of that point.

It is easy to see that the set of (P)-points of a function is thedisjoint union of its left (P)-points, its right (P)-points, andits bilateral (P)-points. What more can we say about these sets?For example, can the set of right (P)-points be uncountable?

Note that the notion of a unilateral (C)-point doesn't arise, sinceTaylor series converge equally far on both sides of their expansionpoint.

To my knowledge, the results in the previous Section have not beeninvestigated for the porosity-typical C-infinity function or for theprevalent C-infinity function (i.e. the complement of a Christensennull set; see).

An investigation for the porosity-typical C-infinity functionwould probably be a good Ph.D. topic, since both the resultsand the proofs will likely be similar to the Baire categoryresults. Also, I've looked into this enough to know that thestrengthening from Baire category to porosity isn't immediate.The Baire category proofs I've seen would have to be restructuredso that you can explicitly get your hands on a decomposition ofnowhere dense sets. After this, you need to have these nowheredense sets sufficiently defined in a constructive way that willallow you to make the necessary porosity computations.

As to whether the prevalent C-infinity function is nowhereanalytic, one way of trying to prove that it is would be tofollow the method of proof given in Christensen [9]. Thiswould require a Fubini/Kuratowski-Ulam type result for Haarnull sets. The exact analog doesn't hold (an example is givenin the paper for MR 48 #4637), but the weaker version that'sgiven in the paper for MR 99g:49013 might suffice.

Other papers that show the existence of a lot of nowhere analyticC-infinity functions are Fabius [14], Kabaya/Iri [17], andFournier/Gauthier [15]. In the first two papers certain distributionfunctions are shown to be C-infinity and nowhere analytic for anysequence of independent and uniformly distributed random variables.The last paper proves that the Baire-typical complex-valued sequencegives rise to a Maclaurin series whose radius of convergence is 0.Results related to this last paper can also be found inSalÃÂÃÂ¡t/TÃÂÃÂ³th [24] and SÃÂÃÂ¡lat/Taylor/TÃÂÃÂ³th [25].

At the very end of the paper Ramsamujh concludes that g does not belong to F(M) from the fact that |g^{(n+1)}(x_0)| > [M^(n+1)]*(n+1)!, where x_0 is a specific point obtained earlier. However, F(M) was defined to be the collection of all C-infinity functions f having the property that there exists an x_0 such that for all n we have |f^{(n+1)}(x_0)| > [M^(n+1)]*(n+1)!. Thus, to show g doesn't belong to F(M) requires a failure at every point x_0 for some n, not the failure at some point x_0 for some n. However, I discussed this problem with Ramsamujh in 1998 and we believe his proof is repairable.